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Pré-Publication, Document De Travail Année : 2019

A note on the neighbour-distinguishing index of digraphs

Résumé

In this note, we introduce and study a new version of neighbour-distinguishing arc-colourings of digraphs. An arc-colouring $\gamma$ of a digraph $D$ is proper if no two arcs with the same head or with the same tail are assigned the same colour. For each vertex $u$ of $D$, we denote by $S_\gamma^-(u)$ and $S_\gamma^+(u)$ the sets of colours that appear on the incoming arcs and on the outgoing arcs of $u$, respectively. An arc colouring $\gamma$ of $D$ is \emph{neighbour-distinguishing} if, for every two adjacent vertices $u$ and $v$ of $D$, the ordered pairs $(S_\gamma^-(u),S_\gamma^+(u))$ and $(S_\gamma^-(v),S_\gamma^+(v))$ are distinct. The neighbour-distinguishing index of $D$ is then the smallest number of colours needed for a neighbour-distinguishing arc-colouring of $D$. We prove upper bounds on the neighbour-distingui\-shing index of various classes of digraphs.
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Dates et versions

hal-02293857 , version 1 (22-09-2019)

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  • HAL Id : hal-02293857 , version 1

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Eric Sopena, Mariusz Woźniak. A note on the neighbour-distinguishing index of digraphs. 2019. ⟨hal-02293857⟩

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